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Let $f:\\mathbb{R}^k\\rightarrow \\mathbb{R}$ be a $C^1$ function, and $\\{z_n\\}$ a sequence constructed from the Bac"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1808.05160","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2018-08-15T15:54:24Z","cross_cats_sorted":["cs.LG","cs.NA","math.NA","stat.ML"],"title_canon_sha256":"bab0e690164ec7fa83dcce2fdbe35b3f29718c605d0d3932e2817e565a8e7a89","abstract_canon_sha256":"ff9d5a8fc88be50ac5524d9645fc94f957a79e591d6d8414b1802ec0d8ad69e4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T02:18:47.837239Z","signature_b64":"lAbszr96/VTxuxtEK8bbmSV1M1Obi9JQMHgNL0mI1ioVTuFK00sdDUsW8L5RJN/G2st8LZRvp00AZjJkUTJZCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"86c5592853d25d5ab8bb076bb237f265e7ffdb3ea851459043484681faf22076","last_reissued_at":"2026-07-05T02:18:47.836835Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T02:18:47.836835Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Backtracking gradient descent method for general $C^1$ functions, with applications to Deep Learning","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.LG","cs.NA","math.NA","stat.ML"],"primary_cat":"math.OC","authors_text":"Tuan Hang Nguyen, Tuyen Trung Truong","submitted_at":"2018-08-15T15:54:24Z","abstract_excerpt":"While Standard gradient descent is one very popular optimisation method, its convergence cannot be proven beyond the class of functions whose gradient is globally Lipschitz continuous. 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Let $f:\\mathbb{R}^k\\rightarrow \\mathbb{R}$ be a $C^1$ function, and $\\{z_n\\}$ a sequence constructed from the Bac"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.05160","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1808.05160/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1808.05160","created_at":"2026-07-05T02:18:47.836894+00:00"},{"alias_kind":"arxiv_version","alias_value":"1808.05160v2","created_at":"2026-07-05T02:18:47.836894+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1808.05160","created_at":"2026-07-05T02:18:47.836894+00:00"},{"alias_kind":"pith_short_12","alias_value":"Q3CVSKCT2JOV","created_at":"2026-07-05T02:18:47.836894+00:00"},{"alias_kind":"pith_short_16","alias_value":"Q3CVSKCT2JOVVOF3","created_at":"2026-07-05T02:18:47.836894+00:00"},{"alias_kind":"pith_short_8","alias_value":"Q3CVSKCT","created_at":"2026-07-05T02:18:47.836894+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2505.22180","citing_title":"Some iterative algorithms on Riemannian manifolds and Banach spaces with good global convergence guarantee","ref_index":47,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/Q3CVSKCT2JOVVOF3A5V3EN7SMX","json":"https://pith.science/pith/Q3CVSKCT2JOVVOF3A5V3EN7SMX.json","graph_json":"https://pith.science/api/pith-number/Q3CVSKCT2JOVVOF3A5V3EN7SMX/graph.json","events_json":"https://pith.science/api/pith-number/Q3CVSKCT2JOVVOF3A5V3EN7SMX/events.json","paper":"https://pith.science/paper/Q3CVSKCT"},"agent_actions":{"view_html":"https://pith.science/pith/Q3CVSKCT2JOVVOF3A5V3EN7SMX","download_json":"https://pith.science/pith/Q3CVSKCT2JOVVOF3A5V3EN7SMX.json","view_paper":"https://pith.science/paper/Q3CVSKCT","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1808.05160&json=true","fetch_graph":"https://pith.science/api/pith-number/Q3CVSKCT2JOVVOF3A5V3EN7SMX/graph.json","fetch_events":"https://pith.science/api/pith-number/Q3CVSKCT2JOVVOF3A5V3EN7SMX/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/Q3CVSKCT2JOVVOF3A5V3EN7SMX/action/timestamp_anchor","attest_storage":"https://pith.science/pith/Q3CVSKCT2JOVVOF3A5V3EN7SMX/action/storage_attestation","attest_author":"https://pith.science/pith/Q3CVSKCT2JOVVOF3A5V3EN7SMX/action/author_attestation","sign_citation":"https://pith.science/pith/Q3CVSKCT2JOVVOF3A5V3EN7SMX/action/citation_signature","submit_replication":"https://pith.science/pith/Q3CVSKCT2JOVVOF3A5V3EN7SMX/action/replication_record"}},"created_at":"2026-07-05T02:18:47.836894+00:00","updated_at":"2026-07-05T02:18:47.836894+00:00"}